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Let $(M,g)$ be a riemannian manifold, $\nabla$ the Levi-Civita connection, and let $X$ be a vector field. I define a positiv metric:

$$h_X=\sum_{i,j} g(\nabla_{e_i}X,\nabla_{e_j}X) [e_i^* \otimes e_j^*]$$

where $(e_i)$ is an orthonormal basis of the tangent space.

$$h_X(Y,Z)=g(\nabla_Y X,\nabla_Z X)$$

If $h_X=g$, then $X$ is called the vector potential of $g$.

Can we have locally a vector potential for any riemannian metric $g$?

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